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Recent Progress in Sphere Packing Recent Progress in Sphere Packing J. H. Conway Mathematics Department Princeton University Princeton, New Jersey 08544 C. Goodman-Strauss Insitituto de Matematicas U.N.A.M. A.P. 273 Admon. de Correos # 3 Cuernavaca, C.P. 62251, Morelos, Mexico N. J. A. Sloane Mathematical Sciences ResearchCenter AT&T Bell Lab oratories Murray Hill, New Jersey 07974 Most of this article will b e ab out comparatively minor progress in the sphere packing problem in higher dimensions, but it is a pleasure to record that at last the original 3-dimensional density problem has b een de nitively solved. The attempted pro of by Hsiang [Hs93] unfortunately turned out to b e incomplete. The Hales Pro of of the Kepler Conjecture for 3-dimensional Sphere Packings The problem of nding the greatest densityofanypacking of equal spheres in n-dimensional Euclidean space is an old and imp ortant one, which has applications in geometry,numb er theory, and information theory. It is also dicult, as is shown by the fact that although the 3-dimensional case was raised by Kepler in 1613, it has only recently b een solved, by Thomas Hales in 1998, with an imp ortantcontribution from S. Ferguson. I shall say only a few words ab out the argument, whichislong and complicated in detail. Hales starts from the familiar Delaunay tessellation, which has one cell for each point of space whose distance from the sphere centers is a lo cal maximum, for which Sloane and I haveintro duced the snappy term \deep hole". Then the Delaunay cell that corresp onds to a deep hole is the convex hull of the sphere centers nearest to that hole, and it is well known that these cells constitute a p olyhedral decomp osition of space. 1 In the standard face-centered cubic lattice packing, the Delaunay cells are alter- nately regular tetrahedra T and o ctahedra O , so that its densityis 0 0 [vol(T )+vol(O )]=[vol(T )+ vol(O )] 0 0 where T and O are the parts of T and O covered by the spheres. However, for a packing in which the sphere centers are in general p osition the Delaunay cells will b e simplicial, and so it suces to establish the density b ound for packings with simplicial Delaunay cells. 0 C. A. Rogers long ago showed that vol(T )=v ol (T )was an upp er b ound for the densityofanypacking|this is nowknown as \the Rogers b ound". It would b e attained for a hyp othetical packing whose Delaunay cells were all regular tetrahedra, so wemaysay that the diculty in the 3-dimensional sphere packing problem is to show that the Rogers b ound must inevitably b e worsened to the extentthatitisin the face-centered cubic packing by the presence of the o ctahedral cells. In the rst pap er of his series, Hales de ned a function called the \score" of a star of Delaunay cells, in terms of another function called the \compression" of a cell. The compression of the Delaunay simplex S is 0 0 vol(S ) vol(S ):v ol (O )=v ol (O ) 0 0 where S is the part of S contained in the spheres, and O and O are as ab ove. In other words, the compression measures howmuchbetter S is covered than is the regular o ctahedral cell O of the face-centered cubic packing. A tetrahedron S is naturally divided into four parts (the Voronoi cells of its vertices), the nth part S consisting of those p oints that are closer to the nth n vertex than any other. In the case that S contains its circumcenter, its score at the nth vertex is de ned to b e its compression, if the circumradius is at most 1:41, and 0 4:sol id(n)=3 4:v ol (S ):v ol (O )=v ol (O ) n otherwise, where sol id(n) denotes the solid angle at that vertex. When S do es not contain its circumcenter, the score is de ned bycontinuing the analytic function corresp onding to this expression. The score (measured in \p oints") of the star of Delaunay tetrahedra for a given vertex is the sum of these scores over all the tetrahedra at that vertex. These de nitions ensure that the average of the score over all of space is the average of the compression, and reduce the problem to proving that the average score is at most 8 p oints. Two sphere centers are called \close neighb ors" if they are distantatmost2.51 from each other, and a Delaunay cell is called a quasi-regular tetrahedron if any two of its four vertices are close neighb ors. In his rst pap er, Hales shows that the score of a star comp osed entirely of quasi-regular tetrahedra is indeed at most 8 points. The remaining pap ers of the sequence are addressed to the much harder task of extending this result to all the other con gurations that mightarise. This involves a combinatorial classi cation of the p ossibilities, whichisinit- self very long, accompanied by an analytical pro of of the appropriate inequality for 2 each of thousands of cases. Both parts involved heavy use of machine computa- tion, and careful selection of various parameters. Hales remarks for instance that \The constant2.51was determined exp erimentally to haveanumb er of desirable prop erties", and similar exp erimental determinations recur rep eatedly throughout the pap er. Several of the initial decisions had to b e mo di ed in the lightoflater calculations. Of course a pro of of this nature is not at all easy to read! But Hales and Ferguson have taken great care to make the entire pro of accessible to readers who wish to checkany detail. In particular they have started a practice that should serveas a mo del for future machine-dep endent pro ofs, bykeeping detailed logs of all their interactions with the machine, so that a p otential auditor can discover that on such and suchaday the following cases were handled, of which the last had to b e split into two sub cases for which the inequalities b ecame ... He or she can then rerun the programs whichwere used to verify the truth of these inequalities. Great care was also taken with these pro ofs. Supp ose, for instance, that the inequality f (x) <g(x) has to b e proved for all x in a certain interval. Then typically the machine will automatically nd a dissection of this interval into a number of subintervals, and nd upp er b ounds for the derivatives of f and g at the endp oints of these which yield linear b ounds for them that still satisfy the inequality.[For higher- dimensional intervals this would involve linear programming.] All calculations of such b ounds are done using interval arithmetic, whichprevents errors that might otherwise arise from rounding the numb ers. 3 Progress in other dimensions: The problem of \b est" packings. In other dimensions, our progress has b een limited, and much of it only conjec- tural. The question we should like to answer is \What are al l the best spherepackings in a given dimension?" Unfortunately, it is not even clear just what this question means! Despite this, the pap er [CS95]ofConway and Sloane gives a conjectural answer to it in each dimension up to 10. We shall roughly follow the discussion in that pap er, marching onward from dimension to dimension, discussing b oth the question and its answers 0 as we go. As the reader maywell susp ect, packings of R are not particularly intriguing, and so we b egin with: 1 What are the b est sphere packings in R ? We can regard a sphere packing by spheres of radius r as a collection of p oints each pair of which is at least r apart. No matter what the b est de nition of \b est" 1 may b e, the b est packing of spheres of radius r in R is surely the lattice of p oints 2kr: And clearly,anypacking that can b e compacted further should not b e called \b est": 1 Note, p erhaps a bit p edantically, that the spheres in a b est packing in R are centered on the p oints of a lattice; we can rescale this to b e the ro ot lattice A , 1 2 that is, the lattice generated bythevector (1; 1) in R . Ro ot lattices will playa large role in our discussion of b est packings; and since the ro ot lattices are canoni- p cally scaled so that the minimum distance b etween p oints is 2, we will adopt the p convention that all of our sphere packings are packings of spheres of radius 2=2. 2 What are the b est sphere packings in R ? Whatever \b est" may mean, the b est packing of the plane should of course b e the hexagonal packing (Figure 1), with circles centered on the p oints of the ro ot lattice A , the lattice generated by the vectors, say,(1; 1; 0) and (0; 1; 1): 2 Clearly \having the highest p ossible density" is a ma jor comp onentofanygood 2 meaning of \b est", and Fejes Toth has given an elegant pro of that no packing in R has density exceeding that of the hexagonal lattice A .However, if weallowthis 2 to b e the entire de nition we get some rather silly \b est packings" since densityis only de ned by a limiting pro cess.
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